Understanding and Reasoning about Multiplication of Fractions

2 What Students Need to Know Well Before Operating With Fractions Meaning of the denominator (number of equal-sized pieces into which the whole has been cut); Meaning of the numerator (how many pieces are being considered); The more pieces a whole is divided into, the smaller the size of the pieces; Fractions aren’t just between zero and one, they live between all the numbers on the number line; A fraction can have many different names; Understand the meanings for operations for whole numbers.

3 A Context for Fraction Multiplication Nadine is baking brownies. In her family, some people like their brownies frosted without nuts, others like them frosted with nuts, and some like them plain. So Nadine frosts 3/4 of the batch of brownies and puts nuts on 2/3 of the frosted part. How much of her batch of brownies has both frosting and nuts?

4 Multiplication of Fractions Consider: How do you think a child might solve each of these? What kinds of reasoning and/or models might they use to make sense of each of these problems?

5 Reasoning About Multiplication Whole number meanings - U.S. conventions 4 x 2 = 8 Set - Four groups of two Area - Four units by two units

6 Reasoning About Multiplication Whole number meanings - U.S. conventions 2 x 4 = 8 Set - Two groups of four Area - Two units by four units When multiplying, each factor refers to something different. One tells how many groups and the other, how many in each group. The representations are quite different.

7 Reasoning About Multiplication Fraction meanings - U.S. conventions Set - Two-thirds of one group of three-fourths Area - Two-thirds of a row of three-fourths of one column Set - Three-fourths of one group of two-thirds Area - Three-fourths rows of two-thirds of one column

8 Models for Reasoning About Multiplication Area/measurement models (e.g., fraction circles) Linear/measurement (e.g., paper tape)

9 Materials for Modeling Multiplication of Fractions How would you use these materials to model ? Paper tape Fraction circles You could also use: Pattern blocks Fraction Bars / Fraction Strips Paper folding

10 Using a Linear Model With Multiplication How much is of ? So of is.

11 Using an Area Model with Fraction Circles for Fraction Multiplication How would you use these materials to model

12 Using a Linear Model With Multiplication How much is of ? So is of 1 is.

13 Using an Area Model with Fraction Circles for Fraction Multiplication How would you use these materials to model ?

14 Mixed Number Multiplication Using a ruler and card, draw a rectangle that is by inches, and find the total number of square inches. Find your answer first by counting, then by multiplying. Compare your answers--are they the same?

15 Mixed Number Multiplication

16 Other Contexts for Multiplication of Fractions Finding part of a part (a reason why multiplication doesn’t always make things “bigger”) Pizza (pepperoni on of a pizza) Ribbon (you have yd of ribbon and need of a yard to make a bow) Lawn ( is mowed, of that is raked)

17 Thinking More Deeply About Multiplication and Division of Fractions Estimating and judging the reasonableness of answers Recognizing situations involving multiplication or division of fractions Considering and creating other contexts where the multiplication of fractions occurs Making thoughtful number choices when considering examples